@stdlib/math/iter/utils/continued-fraction/README.md

Standard math.

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# iterContinuedFraction

> Evaluate the terms of a continued fraction.

<section class="intro">

A **generalized continued fraction** has the form

<!-- <equation class="equation" label="eq:continued_fraction" align="center" raw="x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}}" alt="Continued fraction"> -->

<div class="equation" align="center" data-raw-text="x = b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ldots}}}}" data-equation="eq:continued_fraction">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@d3cef42cbeb8499623814933ed4d906a345cec32/lib/node_modules/@stdlib/math/iter/utils/continued-fraction/docs/img/equation_continued_fraction.svg" alt="Continued fraction">
    <br>
</div> -->

<!-- </equation> -->

If `a_i = 1` for all `i`, the above expression reduces to a **simple continued fraction**.

<!-- <equation class="equation" label="eq:simple_continued_fraction" align="center" raw="x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}}" alt="Simple continued fraction"> -->

<!-- <div class="equation" align="center" data-raw-text="x = b_0 + \frac{1}{b_1+\frac{1}{b_2+\frac{1}{b_3+\frac{1}{b_4+\ldots}}}}" data-equation="eq:simple_continued_fraction">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@d3cef42cbeb8499623814933ed4d906a345cec32/lib/node_modules/@stdlib/math/iter/utils/continued-fraction/docs/img/equation_simple_continued_fraction.svg" alt="Simple continued fraction">
    <br>
</div>

<!-- </equation> -->

where the values `b_i` are called the **coefficients** or **terms** of the continued fraction.

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="usage">

## Usage

```javascript
var iterContinuedFraction = require( '@stdlib/math/iter/utils/continued-fraction' );
```

#### iterContinuedFraction( iterator\[, options] )

Evaluates the terms of a continued fraction.

```javascript
var array2iterator = require( '@stdlib/array/to-iterator' );

var arr = array2iterator( [ 3, 4, 12, 4 ] );

var v = iterContinuedFraction( arr );
// returns ~3.245
```

The function supports the following `options`:

-   **iter**: maximum number of iterations. Default: `1e308`.
-   **tol**: tolerance at which to terminate further evaluation of the continued fraction. Default: [floating-point epsilon][@stdlib/constants/float64/eps].

To explicitly cap the maximum number of iterations, set the `iter` option.

```javascript
var array2iterator = require( '@stdlib/array/to-iterator' );

var arr = array2iterator( [ 3, 4, 12, 4 ] );

var opts = {
    'iter': 2
};
var v = iterContinuedFraction( arr, opts );
// returns 3.25
```

The function terminates continued fraction evaluation once the difference between continued fraction approximations is sufficiently small. The default tolerance is [floating-point epsilon][@stdlib/constants/float64/eps] (`~2.22e-16`). Once an update to a continued fraction approximation is less than or equal to this tolerance, the function terminates evaluation. To adjust the tolerance (e.g., to return a rough approximation), set the `tol` option.

```javascript
var iterContinuedFractionSeq = require( '@stdlib/math/iter/sequences/continued-fraction' );

var opts = {
    'tol': 1.0e-7
};
var it = iterContinuedFractionSeq( 3.141592653589793, opts );
// returns <Object>

var v = iterContinuedFraction( it );
// returns ~3.14159292
```

</section>

<!-- /.usage -->

<!-- Package usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

## Notes

-   If an iterated value is non-numeric (including `NaN`), the function returns `NaN`. If non-numeric iterated values are possible, you are advised to provide an [`iterator`][mdn-iterator-protocol] which type checks and handles non-numeric values accordingly.

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var iterContinuedFractionSeq = require( '@stdlib/math/iter/sequences/continued-fraction' );
var PI = require( '@stdlib/constants/float64/pi' );
var iterContinuedFraction = require( '@stdlib/math/iter/utils/continued-fraction' );

// Create an iterator for returning the list of continued fraction terms:
var it = iterContinuedFractionSeq( PI );

// Reconstitute the original value:
var v = iterContinuedFraction( it );
// returns <number>

console.log( 'Value: %d.', v );
```

</section>

<!-- /.examples -->

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<section class="references">

</section>

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<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/iter/sequences/continued-fraction`][@stdlib/math/iter/sequences/continued-fraction]</span><span class="delimiter">: </span><span class="description">create an iterator which generates a list of all continued fraction terms which can be obtained given the precision of a provided number.</span>

</section>

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<section class="links">

[mdn-iterator-protocol]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Iteration_protocols#The_iterator_protocol

[@stdlib/constants/float64/eps]: https://www.npmjs.com/package/@stdlib/constants-float64-eps

<!-- <related-links> -->

[@stdlib/math/iter/sequences/continued-fraction]: https://github.com/stdlib-js/math/tree/main/iter/sequences/continued-fraction

<!-- </related-links> -->

</section>

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